A certain job was assigned to a group of women with the plan that it would be finished in 20 days. However, 12 women did not turn up for the job, and the remaining women completed the work in 32 days. What was the original number of women in the group that had been planned for the work?

Introduction / Context:
This question deals with a work assignment where the planned number of workers does not all show up. Because fewer women participate, the job takes longer. Using the relationship between the number of workers, time, and total work, we need to find how many women were originally planned for the job.

Given Data / Assumptions:

• Original plan: a group of women was supposed to complete the work in 20 days.
• 12 women did not turn up, so the actual number of women working was reduced by 12.
• The reduced group completed the work in 32 days.
• All women work at the same constant rate and the job size is fixed.

Concept / Approach:
Total work can be expressed as the product of the number of workers, days, and an individual work rate. Since the work is the same in both the planned and actual scenarios, we can equate the expressions for total work. We let the original number of women be N, write total work in terms of N and in terms of N - 12, and solve for N.

Step-by-Step Solution:
Let N be the original number of women planned for the job. Let each woman work at a rate of r units per day, and let the total work be W units. Planned scenario: N women * 20 days * r = W, so W = 20 N r. Actual scenario: only N - 12 women work, and they finish in 32 days. So W = (N - 12) * 32 * r. Equate the two expressions: 20 N r = 32 (N - 12) r. Cancel r on both sides: 20 N = 32 (N - 12). Expand: 20 N = 32 N - 384. Rearrange: 32 N - 20 N = 384, so 12 N = 384. Therefore, N = 384 / 12 = 32 women.

Verification / Alternative check:
Check with actual numbers: planned case has 32 women working 20 days, giving 640 woman days of work. Actual case has 32 - 12 = 20 women working 32 days, again giving 640 woman days of work. Since the total woman days match, our value N = 32 is correct.

Why Other Options Are Wrong:
36 women would give planned work 36 * 20 = 720 woman days, not equal to the actual 20 * 32 = 640 woman days. 22 women or 28 women similarly fail the equality woman days test when 12 are subtracted and time is adjusted. 24 women would result in different total woman days in the two scenarios and cannot be correct.

Common Pitfalls:
Some students try to divide days directly or set up ratios incorrectly without representing the total work explicitly. Another problem is forgetting that 12 women did not turn up, which means the actual number is N - 12, not 12 itself. Always start by letting the original number be N, build equations for total work in both cases, and solve step by step.

Final Answer:
The original number of women in the group was 32.